thresholds for epidemic spreading in networks · 2011. 6. 9. · thresholds for epidemic spreading...
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Thresholds for epidemic spreading
in networksClaudio Castellano
Istituto dei Sistemi Complessi (ISC-CNR), Roma, Italy and
Dipartimento di Fisica, Sapienza Universita’ di Roma, Italy
mercoledì 8 giugno 2011
• Two possible states: susceptible and infected
• Two possible events for infected nodes:
‣ Recovery (rate 1)
‣ Infection to neighbors (rate λ)
Susceptible-Infected-Susceptible(SIS) model
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• Degree distribution P(k) ~ k-γ
• ρk = density of infected nodes of degree k
• Threshold
• Scale-free networks: γ < 3
zero epidemic threshold
HeterogeneousMean-Field theory for SIS
ρk = −ρk + λk[1− ρk]∑
k′
P (k′|k)ρk′
Pastor-Satorras and Vespignani (2001)
λc =〈k〉〈k2〉
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Susceptible-Infected-Removed(SIR) model
• Three possible states: susceptible, infected and removed.
• Two possible events for infected nodes:
‣ Death/recovery (rate 1)
‣ Infection to neighbors (rate λ)
‣ Transition between healthy and infected
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HMF for SIR• HMF theory
• Zero epidemic threshold for scale-free networks
• Finite epidemic threshold for scale-rich networks
λc =〈k〉
〈k2〉 − 〈k〉
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Beyond HMF for SIS
λc =1
ΛN
ΛN =
{c1√
kc√
kc > 〈k2〉〈k〉 ln2(N)
c2〈k2〉〈k〉
〈k2〉〈k〉 >
√kc ln(N)
• Wang et al., 2003
• Chung et al. 2005
ΛN = Largest eigenvalue of the adjacency matrix
kc = Largest degree in the networkmercoledì 8 giugno 2011
Beyond HMF for SIS• Summing up
• In any uncorrelated quenched random network with power-law distributed connectivities, the epidemic threshold goes to zero as the system size goes to infinity.
• This has nothing to do with the scale-free nature of the degree distribution.
λc !{
1/√
kc γ > 5/2〈k〉〈k2〉 2 < γ < 5/2
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Finite Size ScalingSIS γ = 4.5
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Mathematical originof HMF failure for SIS
• HMF is equivalent to using annealed networks with adjacency matrix
• This matrix has a unique nonzero eigenvalue
ΛN =〈k2〉〈k〉
aij → a(ki, kj) =kikj
〈k〉N
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Physical originof HMF failure for SIS
• Star graph with kmax+1 nodes
• For the hub and its neighbors are a self-sustained core of infected nodes, which spreads the activity to the rest of the system.
ρmax ∝ (λ2kmax − 1)
ρ1 ∝ (λ2kmax − 1)
λ > 1/√
kmax
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A truly endemic stateγ = 3.5
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Extensions
• For Erdos-Renyi graphs
• For correlated graphs
ΛN → max{√
kmax, 〈k〉}
Krivelevich et al., 2003
ΛN ≥√
kmax
Restrepo et al, 2007
The threshold vanishes for any network
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SIR γ = 4.5
Conjecture: on scale-rich networks, the epidemic threshold is vanishing or finite depending on the presence or absence
of a steady-state.mercoledì 8 giugno 2011
Kitsak et al., Nat. Phys. (2010)
k-cores are important for SIR (and SIS...)mercoledì 8 giugno 2011
101 102 103 104qmax100
101
102
kS<qkS
>
qkSmax
SIS dynamics onmaximum k-core
•The maximum k-core has a narrow degree distribution
λc ≈1〈k〉 ∼
1kS
γ = 2.5
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SIS dynamics onmaximum k-core
• Maximum k-core index (Dorogovtsev et al, 2007)
• Summing up
The maximum k-core induces a thresholdscaling as the HMF threshold
λc ∼ 1/kS ∼ kγ−3max ∼ 〈k〉/〈k2〉
kS ≈ (γ − 2)(3− γ)(3−γ)/(γ−2)kmax
(kmin
kmax
)(γ−2)
∼ k3−γmax
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1 10 100t
1e-08
1e-06
0,0001
0,01
1ρ
λ = 0.01λ = 0.02λ = 0.03
UCMγ = 2.75
1 10 100t
1e-05
0,0001
0,001
0,01
0,1
1
ρ
λ = 0.01λ = 0.02λ = 0.03
maximum k-coreγ = 2.75
1 10 100t
1e-06
0,0001
0,01
1
ρ
λ = 0.01λ = 0.02λ = 0.03
Star graphkmax = 5462
Transition governedby the hub
1/ΛN = 0.0131
1/√
kmax = 0.0135
〈k〉/〈k2〉 = 0.0231
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1 10 100t
1e-08
1e-06
0,0001
0,01
1
ρ
λ = 0.005λ = 0.01λ = 0.02λ = 0.03
UCMγ = 2.1
1 10 100t
1e-08
1e-06
0,0001
0,01
1
ρ
λ = 0.005λ = 0.01λ = 0.02λ = 0.03
star graphkmax = 999
1 10 100t
1e-08
1e-06
0,0001
0,01
1
ρ
λ = 0.005λ = 0.01λ = 0.02λ = 0.03
maximum k-coreγ = 2.1
Transition governedby the maximum k-core
1/ΛN = 0.00735
1/√
kmax = 0.03163
〈k〉/〈k2〉 = 0.00745
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SIR dynamics• On the maximum k-core
• On the star-graph centered around the hub
The maximum k-core always governsthe transition and sets the threshold
λc ∼ 1/kS ∼ kγ−3max ∼ 〈k〉/〈k2〉
No dependence on kmax (when large)
R = λ2 +1 + 2λ
kmax
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10-3 10-2 10-110-6
10-4
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R
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!
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10-2 10-110-6
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100
γ = 2.75
λ
R
γ = 2.25
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Movies
PGP
AS
Network max k-core Hub
t
ρ
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Summary• Zero epidemic threshold for SIS on scale-rich networks.
• Finite epidemic threshold for SIR on scale-rich networks.
• For SIS the transition can be governed either by the maximum k-core or by the largest hub.
• For SIR the maximum k-core always dominates.
• Correlations may change the picture
C. Castellano and R. Pastor-Satorras, PRL, 105, 218701 (2010)C. Castellano and R. Pastor-Satorras, arXiv:1105.5545 (2011)
mercoledì 8 giugno 2011